There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars---scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

Culture shapes the objects people use and for what purposes, yet mainstream Vision-Language (VL) datasets frequently exhibit cultural biases, disproportionately favoring higher-income, Western contexts. This imbalance reduces model generalizability and perpetuates performance disparities, especially impacting lower-income and non-Western communities. To address these disparities, we propose a novel function-centric framework that categorizes objects by the functions they fulfill, across diverse cultural and economic contexts. We implement this framework by creating the Culture Affordance Atlas, a re-annotated and culturally grounded restructuring of the Dollar Street dataset spanning 46 functions and 288 objects publicly available at https://lit.eecs.umich.edu/CultureAffordance-Atlas/index.html. Through extensive empirical analyses using the CLIP model, we demonstrate that function-centric labels substantially reduce socioeconomic performance gaps between high- and low-income groups by a median of 6 pp (statistically significant), improving model effectiveness for lower-income contexts. Furthermore, our analyses reveals numerous culturally essential objects that are frequently overlooked in prominent VL datasets. Our contributions offer a scalable pathway toward building inclusive VL datasets and equitable AI systems.

Here we introduce methods to extend the approach to models with higher-order potentials and develop theoretical insights. In particular, we show that the triplet-consistent polytope TRI is unique in being'universally rooted'.

Stigmatizing language results in healthcare inequities, yet there is no universally accepted or standardized lexicon defining which words, terms, or phrases constitute stigmatizing language in healthcare. We conducted a systematic search of the literature to identify existing stigmatizing language lexicons and then analyzed them comparatively to examine: 1) similarities and discrepancies between these lexicons, and 2) the distribution of positive, negative, or neutral terms based on an established sentiment dataset. Our search identified four lexicons. The analysis results revealed moderate semantic similarity among them, and that most stigmatizing terms are related to judgmental expressions by clinicians to describe perceived negative behaviors. Sentiment analysis showed a predominant proportion of negatively classified terms, though variations exist across lexicons. Our findings underscore the need for a standardized lexicon and highlight challenges in defining stigmatizing language in clinical texts.

We address the problem of verifying automatically procedural programs manipulating parametric-size arrays of integers, encoded as a constrained Horn clauses solving problem. We propose a new algorithmic method for synthesizing loop invariants and procedure pre/post-conditions represented as universally quantified first-order formulas constraining the array elements and program variables. We adopt a data-driven approach that extends the decision tree Horn-ICE framework to handle arrays. We provide a powerful learning technique based on reducing a complex classification problem of vectors of integer arrays to a simpler classification problem of vectors of integers . The obtained classifier is generalized to get universally quantified invariants and procedure pre/post-conditions. We have implemented our method and shown its efficiency and competitiveness w.r.t.

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality d . The key observation is that nonlinear O( d)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars---scalar products and scalar contractions of the scalar, vector, and tensor inputs.

Reward machines (RMs) are an effective approach for addressing non-Markovian rewards in reinforcement learning (RL) through finite-state machines. Traditional RMs, which label edges with propositional logic formulae, inherit the limited expressivity of propositional logic. This limitation hinders the learnability and transferability of RMs since complex tasks will require numerous states and edges. To overcome these challenges, we propose First-Order Reward Machines ($\texttt{FORM}$s), which use first-order logic to label edges, resulting in more compact and transferable RMs. We introduce a novel method for $\textbf{learning}$ $\texttt{FORM}$s and a multi-agent formulation for $\textbf{exploiting}$ them and facilitate their transferability, where multiple agents collaboratively learn policies for a shared $\texttt{FORM}$. Our experimental results demonstrate the scalability of $\texttt{FORM}$s with respect to traditional RMs. Specifically, we show that $\texttt{FORM}$s can be effectively learnt for tasks where traditional RM learning approaches fail. We also show significant improvements in learning speed and task transferability thanks to the multi-agent learning framework and the abstraction provided by the first-order language.

The idea of uprooting and rerooting graphical models was introduced specifically for binary pairwise models by Weller [19] as a way to transform a model to any of a whole equivalence class of related models, such that inference on any one model yields inference results for all others. This is very helpful since inference, or relevant bounds, may be much easier to obtain or more accurate for some model in the class. Here we introduce methods to extend the approach to models with higher-order potentials and develop theoretical insights. In particular, we show that the triplet-consistent polytope TRI is unique in being'universally rooted'. We demonstrate empirically that rerooting can significantly improve accuracy of methods of inference for higher-order models at negligible computational cost.